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Journal of the Optical Society of America B

Competition between Zeeman and hyperfine pumping in thermal atomic vapor

Rajni Bala, Joyee Ghosh, Vivek Venkataraman

Published Aug 29, 2023• 1 citation
Publisher

Abstract

We present ab initio reduced rate equation models to study the interplay of Zeeman and hyperfine pumping, which determines the Doppler-broadened absorption spectra of an arbitrarily polarized probe laser, in thermal atomic vapor. Our simple models (with maximum seven levels) accurately predict the dependence of probe absorption on its polarization for the 87Rb D2 line (residual error <2%) without any fitting parameters, while being ∼100× less computational resource intensive as compared to numerically complex full multi-level models that treat all (up to 23) the magnetic (m F ) sub-levels present in the system individually. We demonstrate that either Zeeman or hyperfine pumping dominates depending on whether the atomic transitions being probed are primarily closed or open.

DOI
10.1364/josab.498324

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We present ab initio reduced rate equation models to study the interplay of Zeeman and hyperfine pumping, which determines the Doppler-broadened absorption spectra of an arbitrarily polarized probe laser, in thermal atomic vapor. Our simple models (with maximum seven levels) accurately predict the dependence of probe absorption on its polarization for the 87Rb D2 line (residual error <2%) without any fitting parameters, while being ∼100× less computational resource intensive as compared to numerically complex full multi-level models that treat all (up to 23) the magnetic (m F ) sub-levels present in the system individually. We demonstrate that either Zeeman or hyperfine pumping dominates depending on whether the atomic transitions being probed are primarily closed or open.
Abstract We theoretically and experimentally investigate the Doppler-broadened absorption spectrum of the D2-line (52 S 1/2 → 52 P 3/2) in thermal (room-temperature) Rb vapour over a wide range of probe intensity, probe diameter, and vapour cell temperature to understand the effect of hyperfine pumping and transit-relaxation on the absorption lineshape. We present a relatively simple but comprehensive five-level rate equation model which incorporates optical pumping of the atomic population into the other ‘dark’ ground hyperfine level (leading to absorption saturation), including contributions of the three closely spaced (within one Doppler linewidth) hyperfine levels in the excited state 52 P 3/2 (allowed by the dipole transition selection rules). D2-line transmission spectra predicted by our model show excellent agreement (rms error < 5 % ) with the experimental data for a wide range of probe intensities (from ∼ 0.001 I s a t 0 all the way up to ∼ 10 I s a t 0 , where I sat0 is the usual two-level saturation intensity for the respective atomic transition), vapour cell temperatures (24.2°C–53.5 °C) and beam diameters (1/e 2 width ∼ 1 − 6 mm). Our model also takes into account the finite transit time of atoms across the probe beam cross-section, and correctly predicts the dependence of effective saturation intensity on probe diameter (up to ∼ 10 × lower saturation intensity as compared to I sat0 for few-mm beam sizes). This ab initio five-level model can thus predict accurate Doppler lineshapes for any given experimental parameter set for a linearly polarized probe without any fitting parameters, and can be easily applied to any other atomic system by an appropriate change of the atomic constants in the rate equations.
We study the pump and probe absorption spectra, as a function of the probe detuning, in a degenerate two-level atomic system, for the case where the probe intensity is high enough to affect the pump absorption. The theory is valid for any ${F}_{g}\ensuremath{\rightarrow}{F}_{e}$ alkali-metal transition interacting with an arbitrarily intense pump and probe (with general Rabi frequencies ${\ensuremath{\Omega}}_{1,2}$) which are perpendicularly polarized with either ${\ensuremath{\sigma}}_{\ifmmode\pm\else\textpm\fi{}}$ or $\ensuremath{\pi}$ polarization. We have constructed a computer program that can calculate the spectra without requiring one to write out the Bloch equations explicitly. We show that, when the pump is ${\ensuremath{\sigma}}_{\ifmmode\pm\else\textpm\fi{}}$ polarized and the probe $\ensuremath{\pi}$ polarized, or vice versa, the pump and probe absorptions depend on the Zeeman coherences between the nearest-neighboring ground or excited Zeeman sublevels, whereas when the pump is ${\ensuremath{\sigma}}_{+}$ polarized and the probe ${\ensuremath{\sigma}}_{\ensuremath{-}}$ polarized, or vice versa, the Zeeman coherences that directly determine the absorption are between next-nearest neighbors. We report calculations of the pump and probe absorption spectra for the cycling ${F}_{g}=2\ensuremath{\rightarrow}{F}_{e}=3$ transition in the D2{D}_{2}D2​ line of 87Rb^{87}\text{R}\text{b}87Rb, interacting with a resonant ${\ensuremath{\sigma}}_{+}$-polarized pump and either a $\ensuremath{\pi}$- or a ${\ensuremath{\sigma}}_{\ensuremath{-}}$- polarized probe. The probe and pump absorption spectra are analyzed by considering the contributions that derive from the individual ${m}_{g}\ensuremath{\rightarrow}{m}_{e}$ transitions. We then show how these contributions depend on the ground- and excited-state populations and Zeeman coherences, and investigate the role played by transfer of coherence from the excited to the ground hyperfine state. We show that the pump and probe absorption spectra are mirror images of each other when ${\ensuremath{\Omega}}_{1}\ensuremath{\ge}{\ensuremath{\Omega}}_{2}>\ensuremath{\Gamma}$, and have the same behavior at line center and complementary behavior in the wings, when ${\ensuremath{\Omega}}_{1}\ensuremath{\ge}{\ensuremath{\Omega}}_{2}<\ensuremath{\Gamma}$ ($\ensuremath{\Gamma}$ is the rate of spontaneous decay from Fe{F}_{e}Fe​ to Fg{F}_{g}Fg​).
We study the Doppler-broadened absorption of a weak monochromatic probe beam in a thermal rubidium vapour cell on the D lines. A detailed model of the susceptibility is developed which takes into account the absolute linestrengths of the allowed electric dipole transitions and the motion of the atoms parallel to the probe beam. All transitions from both hyperfine levels of the ground term of both isotopes are incorporated. The absorption and refractive index as a function of frequency are expressed in terms of the complementary error function. The absolute absorption profiles are compared with experiment, and are found to be in excellent agreement provided a sufficiently weak probe beam with an intensity under one thousandth of the saturation intensity is used. The importance of hyperfine pumping for open transitions is discussed in the context of achieving the weak-probe limit. Theory and experiment show excellent agreement, with an rms error better than 0.2% for the D2 line at 16.5 degrees C.
Abstract. We study the Doppler-broadened absorption of a weak monochromatic probe beam in a thermal rubidium vapour cell on the D lines. A detailed model of the susceptibility is developed which takes into account the absolute linestrengths of the allowed electric dipole transitions and the motion of the atoms parallel to the probe beam. All transitions from both hyperfine levels of the ground term of both isotopes are incorporated. The absorption and refractive index as a function of frequency are expressed in terms of the complementary error function. The absolute absorption profiles are compared with experiment, and are found to be in excellent agreement provided a sufficiently weak probe beam with an intensity under one thousandth of the saturation intensity is used. The importance of hyperfine pumping for open transitions is discussed in the context of achieving the weak-probe limit. Theory and experiment show excellent agreement, with an rms error better than 0.2 % for the D2 line at 16.5 ◦ C. PACS numbers: 32.70.-n, 32.70.Fw, 42.62.FiAbsolute Rb absorption 2 1.
We propose and demonstrate pump-probe spectroscopy of rubidium absorption that reveals the sub-Doppler hyperfine structure of the ${}^{5}{\mathrm{S}}_{1/2}\ensuremath{\leftrightarrow}{}^{5}{\mathrm{P}}_{3/2}$ (D2{\mathrm{D}}_{2}D2​) transitions. The counterpropagating pump and probe lasers are independently tunable in frequency, with the probe operating at the single-photon level. The two-dimensional spectrum measured as the laser frequencies are scanned shows fluorescence, Doppler-broadened absorption dips, and sub-Doppler features. The detuning between the pump and probe lasers allows compensation of the Doppler shift for all atomic velocities in the room-temperature vapor, meaning we observe sub-Doppler features for all atoms in the beam. We detail a theoretical model of the system that incorporates fluorescence, saturation effects, and optical pumping and compare this with the measured spectrum, finding a mean absolute percentage error of 3.11%. This high level of agreement allows us to extract the number density and temperature of the Rb\mathrm{Rb}Rb vapor from the two-dimensional spectrum. In the future this technique could be used as an atomic frequency standard for calibrating and characterising a single-photon or low-light-level source.